scheidt and caers (2010)

8/19/2019 Scheidt and Caers (2010)

1/23

Math Geosci (2009) 41: 397–419

DOI 10.1007/s11004-008-9186-0

Representing Spatial Uncertainty Using Distances

and Kernels

Céline Scheidt · Jef Caers

Received: 2 July 2007 / Accepted: 17 June 2008 / Published online: 24 September 2008

© International Association for Mathematical Geology 2008

Abstract Assessing uncertainty of a spatial phenomenon requires the analysis of a

large number of parameters which must be processed by a transfer function. To cap-

ture the possibly of a wide range of uncertainty in the transfer function response, a

large set of geostatistical model realizations needs to be processed. Stochastic spa-

tial simulation can rapidly provide multiple, equally probable realizations. However,

since the transfer function is often computationally demanding, only a small num-

ber of models can be evaluated in practice, and are usually selected through a rank-ing procedure. Traditional ranking techniques for selection of probabilistic ranges

of response (P10, P50 and P90) are highly dependent on the static property used.

In this paper, we propose to parameterize the spatial uncertainty represented by a

large set of geostatistical realizations through a distance function measuring “dissim-

ilarity” between any two geostatistical realizations. The distance function allows a

mapping of the space of uncertainty. The distance can be tailored to the particular

problem. The multi-dimensional space of uncertainty can be modeled using kernel

techniques, such as kernel principal component analysis (KPCA) or kernel clustering.

These tools allow for the selection of a subset of representative realizations containingsimilar properties to the larger set. Without losing accuracy, decisions and strategies

can then be performed applying a transfer function on the subset without the need

to exhaustively evaluate each realization. This method is applied to a synthetic oil

reservoir, where spatial uncertainty of channel facies is modeled through multiple re-

alizations generated using a multi-point geostatistical algorithm and several training

images.

C. Scheidt () · J. CaersDepartment of Energy Resources Engineering, Stanford University, 367 Panama Street, Green Earth

Sciences 353, Stanford, CA 94305-2220, USA

e-mail: [email protected]

J. Caers

e-mail: [email protected]

mailto:[email protected]:[email protected]:[email protected]:[email protected]


2/23

398 Math Geosci (2009) 41: 397–419

Keywords Uncertainty quantification · Distance · Kernel methods · Ranking ·

Geostatistics

1 Introduction

Stochastic spatial simulation is now widely used to generate multiple, alternative re-

alizations or samples of the same underlying spatial phenomenon, representing the

uncertainty of the simulated variable(s). This set of realizations models the so-called

“space of uncertainty” of the underlying phenomenon. In most applications, these re-

alizations are not sufficient to assess uncertainty; further processing must be applied

to address the practical questions at hand (Journel and Alabert 1990). For example, in

reservoir engineering, several realizations of petrophysical and/or lithological mod-

els of the subsurface reservoir are generated and then submitted to flow simulationto assess reservoir flow performance, to assess the impact of drilling new wells or to

optimize their placement. A similar situation arises in the management of groundwa-

ter, where subsurface model realizations are used to assess the impact of pumping on

groundwater flow.

In general, a “transfer function” (e.g., flow simulator) is applied to post-process

each realization and thereby obtain its “response” (e.g., reservoir performance),

which may be single-valued or consist of a time-varying response. If many realiza-

tions are processed through the same transfer function, a probability distribution of

the response can be constructed and serve as a model of uncertainty (Fig. 1). Whilethis Monte Carlo simulation framework appears general and straightforward, sev-

eral challenges make it difficult to apply. First, the set of realizations may be gener-

ated by varying several key parameters impacting spatial variation. Applying a single

Fig. 1 Schematic diagram showing the calculation of probability quantiles though the evaluation of many

realizations using a transfer function


3/23

Math Geosci (2009) 41: 397–419 399

geostatistical algorithm with a fixed parameter input often does not cover a wide

enough space of uncertainty. However, by jointly varying several input parameters

(variogram, histogram, training image, etc.) or even by varying the generating algo-

rithm itself, one may need to create several hundred or thousand of realizations to

capture the possible space of uncertainty adequately. Second, the transfer functionmay be expensive to evaluate in terms of CPU. Many transfer functions are either

finite difference or finite element codes that may require some form of iterative op-

timization which may take several CPU hours if the grid underlying the realizations

contains a large (105, 106) number of cells. It is therefore impractical, in most cases,

to process several hundreds or thousand of realizations.

To circumvent these challenges, a widely used approach for modeling uncertainty

is the experimental design technique (Box and Draper 1975). Experimental design

aims at optimally selecting values of uncertain parameters in their range of varia-

tion, and then applying the transfer function on the resulting realizations to obtainresponses (Damslet et al. 1992; Manceau et al. 2001). From the response values, a

proxy model of the transfer function is built, which is a function of the uncertain pa-

rameters. The proxy model permits traditional Monte Carlo analysis of uncertainty.

However, one major drawback of experimental design techniques is that they are

often based on a simple linear regression and are thus not well suited for spatial vari-

ables or high-dimensional problems. In addition, experimental design techniques are

not appropriate for applications with many discrete parameters.

An alternative approach to quantifying uncertainty is to examine a large set of real-

izations, and not individual parameters as done in experimental design. For problemsof large dimensionality, evaluating the uncertainty of each parameter separately is of-

ten not useful, since many parameters are correlated, frequently in complex fashions.

Moreover, ultimately, we are not interested in uncertainty of individual parameters,

but in the realizations built from these parameters and responses predicted from those

realizations. Ranking techniques are traditionally used to select realizations that rep-

resent the P10, P50 and P90 quantiles of the responses of interest (Ballin et al. 1992).

The kth quantile is defined as the value x such that the probability of the response will

be less than x is at most k% and the probability that the response will be less than

or equal to x is at least k%. However, ranking techniques are highly dependent on

the ranking property employed. Ranking is often based on a rather simple statistics

extracted from the realization (e.g., original oil-in-place for reservoir models), which

may not correctly capture the transfer function behavior. These statistical measure-

ments often have a poor correlation with the response measured from the transfer

function.

In this paper we propose a new approach which is well suited to treat problems

using a very complex, time consuming transfer function, and non-Gaussian fields,

which do not fit with traditional Least-Squares techniques. The key concept is the

construction of a realization-based representation of uncertainty parameterized by

distances. This method employs a single parameter (the distance) between any two

realizations, which can be tailored to a particular application at hand. The aim is to

select a set of representative realizations by analyzing the properties of the realiza-

tions as characterized by the distance, and finally quantify uncertainty within that

set. Contrary to experimental design methodology which constructs a proxy model


4/23

400 Math Geosci (2009) 41: 397–419

Fig. 2 Estimation of quantiles P10, P50 and P90 of a large set of responses by using only a few well

selected realizations

of the response, the proposed method relies on the assumption that a few selected

realizations have the same statistical characteristics in terms of response as the en-tire set (Fig. 2). Thus, no prediction outside of the existing set of responses is done

with this method—the key is to select properly the subset of realizations. The follow-

ing section describes the methodology used in this approach. An application of the

methodology on a synthetic reservoir case of channel facies is presented. We end this

paper by giving some conclusions and a discussion of future work.

2 Description of Methodology

The objective of the methodology is to efficiently select, among a potentially large

set of realizations, a subset whose response (evaluated from a given transfer function)

exhibits the same statistical properties (densities, quantiles, mean, variance, etc.) as

the entire set of realizations. Since the transfer function can be very CPU demand-

ing, the objective is to avoid evaluating realizations having similar responses, and to


5/23

Math Geosci (2009) 41: 397–419 401

concentrate on realizations which span a variety of response behavior. One way to do

so is to attempt to quantify differences or similarities between realizations, and then

group together realizations which are similar.

The principle of the methodology, illustrated for realizations of a binary categor-

ical variable, is described in Fig. 3. Each step in Fig. 3 will be described in greaterdetail below. Starting with multiple (N R) realizations generated using any algorithm,

a dissimilarity distance matrix is constructed (Fig. 3(A) and 3(B)). This N R ×N R ma-

trix contains the “distance” between any two realizations. The matrix is then used to

map all realizations into a Euclidean spaceR (Fig. 3(C)) using multidimensional scal-

ing. Each point in this map represents a realization. Since in most cases the structure

of the points in mapping space R is not linear, we use kernel methods to transform

the Euclidean space R into a new space F , called the feature space (Fig. 3(D)). The

goal of the kernel transform is that points in this new space behave more linearly, so

that standard linear tools for pattern detection can be used more successfully (Prin-cipal Component Analysis, cluster analysis, dimensionality reduction, etc.). These

tools allow the selection of a few “typical” points representing realizations, among

a potentially very large set. Application of the transfer function on a small subset of

realizations allows uncertainty quantification (e.g., P10, P50, P90 quantiles) of the

response variable.

2.1 Measurement of Dissimilarity Distance

2.1.1 Definition of Distance

The first step of the methodology is the definition of a dissimilarity distance between

any two realizations (Fig. 3(A)). The concept of similarity between geostatistical

model realizations was introduced by Arpat (2005), and Suzuki and Caers (2008).

The distance is a way to determine how similar two realizations are in terms of spa-

tial properties and transfer function response. The distance between two realizations

can be determined by classical distances that measure difference in geometry such

as the Hausdorff distance (Dubuisson and Jain 1994). For simulation of categori-cal variables, the Hausdorff distance focuses on foreground facies pixels (if binary

model) or edge pixels of binary edges extracted from any type of realizations (Suzuki

and Caers 2008). However, the Hausdorff distance does not take into account con-

nectivity between wells which may be necessary to evaluate properly differences in

flow. Connectivity-based distances (Park and Caers 2007), or streamline-based sim-

ulation are other examples of possible distances. Note that the concept of “relative”

distance between two “objects” is different from the concept of difference in “ab-

solute” statistical summaries. It is not necessary to determine statistical measures

for each realization to measure dissimilarity. We need only to define a distance be-

tween any two realizations. The dissimilarity distance can be measured in any fash-

ion; the only requirement for the distance between two realizations is that it should

have a reasonable correlation with the difference in response of the same two realiza-

tions.


6/23

402 Math Geosci (2009) 41: 397–419

Fig. 3 Proposed workflow for uncertainty quantification—(A) distance between two models, (B) distance

matrix D, (C) models mapped in Euclidean space, (D) feature space, (E) pre-image construction, (F) P10,

P50 and P90 quantile estimations


7/23

Math Geosci (2009) 41: 397–419 403

2.1.2 Construction of a Dissimilarity Distance Matrix

Given a set of N R realizations yi and a distance function δ between any two real-

izations, a N R × N R dissimilarity distance matrix D is constructed containing the

distance measured between any two realizations δij . A valid dissimilarity matrix

must satisfy both of the following constraints: self-similarity (δii = 0) and symmetry

(δij = δj i ). Once the distance matrix D is constructed, all the N R realizations are

mapped into an Euclidean space R using multidimensional scaling (MDS).

2.2 Multidimensional Scaling (MDS)

MDS is a technique used to translate the dissimilarity matrix into a configuration of

points in nD Euclidean space (Borg and Groenen 1997; Cox and Cox 1994). The

points in this spatial representation are arranged in such a way that their Euclidean

distances correspond as much as possible (in least square sense) to the dissimilarities

of the objects. A successful MDS procedure results in a good correlation between the

Euclidean distance and the dissimilarity distance. The classical MDS algorithm rests

on the fact that the coordinate matrix X of the points can be derived by eigenvalue

decomposition from a matrix A obtained by converting the dissimilarity matrix D

into a scalar product. Note that since the map obtained by MDS is derived solely

by the dissimilarity distances in the matrix, the absolute location of the points isirrelevant. The map can be subject to translation, rotation, and reflection, without

impacting the methodology. Only the distances in mapping space R are of interest.

Applying this concept to our methodology, the objects under consideration are

realizations of a spatial phenomena, thus each realization is represented as a point.

MDS allows to represent each realization yi ∈ RN C , i = 1, . . . , N R (defined by poten-

tially millions of grid-blocks) in a reduced coordinate system xi ∈ Rp, i = 1, . . . , N R .

The dimension p of the Euclidean space is defined according to the eigenvalue de-

composition of A. In most cases, p can be small, since only a few eigenvalues in

the decomposition are significant. A higher dimensional space could be used to in-crease the correlation, however, it would be improved only slightly while making

the convergence of the pre-image more difficult (see below for the description of the

pre-image).

An illustration of an application of classical MDS is presented in Fig. 4, illustrated

for facies models and using the Hausdorff distance to construct the dissimilarity ma-

trix. In this example, a 2D Euclidean space is sufficient to obtain a good quality map-

ping. The Euclidean distances between any two points are very close to the distances

in the dissimilarity matrix (correlation of 0.79). Figure 5 shows the increase of the

correlation as a function of the dimension of the Euclidean space. Note that the use

of high dimensional Euclidean spaces may not increase the correlation significantly.

For most applications, the structure of the points in the mapping space R is not linear

and therefore standard tools for pattern detection are not appropriate. To avoid this

problem, we employ kernel methods.


8/23

404 Math Geosci (2009) 41: 397–419

Fig. 4 Multidimensional Scaling (MDS): each point represents a reservoir model in a 2D space

Fig. 5 Correlation between

dissimilarity distance and

Euclidean distance as a function

of the dimension of the

Euclidean space


9/23

Math Geosci (2009) 41: 397–419 405

2.3 Kernel Methodology

2.3.1 Kernel-Principle

Kernel theory was recently developed in the field of neural computing and patternrecognition (Vapnick 1998). Kernel principal component analysis (KPCA) is often

used as a tool to remove noise from computerized images (Shawe-Taylor and Cris-

tianni 2004). In reservoir engineering, kernel theory has been used by Sarma (2006)

in the context of inversion of flow data and production optimization. Kernel methods

consist of mapping the given data points from their input space R to some high-

dimensional feature space F using a multidimensional function Φ : Φ : R → F . The

feature space F is assumed to have a better linear variation than R. In other words,

points in F are linearly separable. Thus, tools requiring a linear relationship between

data can be applied into F instead of R. In our application, points generated by MDSin space R are transformed by kernel methods into space F .

Kernel methods can be used to develop nonlinear generalizations of any algo-

rithm that can be cast in term of scalar products, such as PCA or k-means clustering

(Schöelkopf et al. 1998; Schöelkopf and Smola 2002). One principle advantage of

using kernels in these applications is that there is no need to map explicitly the points

from space R to F ; all necessary computations in space F can be carried out us-

ing the scalar product of the nonlinear function Φ. This function is called a kernel

function k , and is given by

k(x, y) =Φ(x),Φ(y)

. (1)

The most common kernel function, the scalar product in the feature space F , is the

Gaussian kernel (radial basis function), which is given by

k(x, y) = exp

−x − y2

2σ 2

with σ > 0. (2)

In our application, we consider a Gaussian kernel for all the cases (2). Tests on other

kernel functions such as polynomial or sigmoidal were made, but the Gaussian kernelwas found to be more robust. The kernel width parameter σ controls the flexibility

of the kernel. From Keerthi and Lin (2003), we know that for small values of σ , the

kernel matrix becomes close to identity matrix (K = I ), and thus the application of

KPCA will lead to severe over-fitting. On the other hand, large values of σ gradually

reduce the kernel to a constant function (K = 1). In this case, the application of

KPCA will lead to severe under-fitting. As recommended by Shi and Malik (2000),

we choose σ as 10% to 20% of the range of the distance between points. Robustness

in the results of kernel clustering were found using σ within this range.

2.3.2 Pre-Image Construction

While the mapping Φ from input space R to feature space F is of primary importance

in kernel methods, the reverse mapping from feature space F back to input space R

may be desired (Fig. 3(E)). This reverse mapping process is called the pre-image


10/23

406 Math Geosci (2009) 41: 397–419

problem. For example, one may want to map back to R the points (in space F )

projected by KPCA into a lower dimensional space. The difficulty with this procedure

is that the mapping function Φ into F is unknown, nonlinear and non-unique, thus

only approximate solutions are possible. In this work, approximate pre-images are

found using the fixed-point iteration approach proposed by Schölkopf. This approachis essentially a gradient-based optimization technique. Note that the pre-image points

do not necessarily correspond to a point in the original space R. In this case, we take

the closest existing point. For details about the pre-images algorithm, see Schöelkopf

and Smola (2002).

2.3.3 Kernel Principal Component Analysis (KPCA)

To understand KPCA better, we first recall quickly the theory of principal component

analysis (PCA). PCA consists of projecting linearly the data onto a lower-dimensional

space. PCA provides a set of orthogonal axes, called principal components, obtained

by solving the eigenvalue problem of the sample correlation matrix. A small number

of principal components is often sufficient to describe the major trend in the data.

KPCA works in a similar manner. First, kernels map the given data points from space

R to space F using a multidimensional function Φ : Φ : R→ F and then PCA is

applied in F . Using the kernel function (1), it can be shown (Schöelkopf and Smola

2002) that KPCA requires only an eigenvalue decomposition of the N R × N R kernel

matrix K defined by

Kij = k(xi , xj ), xi ∈ R, i = 1, . . . , N R.

KPCA is a powerful technique for extracting structure from potentially high-

dimensional data sets with complex variability. KPCA can be seen as a way of re-

moving noise from the points in R.

2.3.4 Kernel K-Means Clustering (KKM)

Clustering algorithms are also applicable in feature space F and are suited to our

problem. Cluster analysis aims to discover the internal organization of a dataset by

finding structure within the data in the form of clusters. Hence, the data is broken

down into a number of groups composed of similar objects. This methodology is

widely used both in multivariate statistical analysis and in machine learning. Defin-

ing clusters consists in identifying an ‘a priori’ fixed number of centers and assign

points to clusters with the closest center. The number of the cluster is defined by the

user, depending on the number of evaluations of the transfer function which can be

performed in the time allotted to the task. In this work, we apply the classical k-means

algorithm in the feature space F to determine a subset of points defined by the clus-

ter centroids. The k-means procedure requires a method for measuring the distance

between two points in the high-dimensional feature space F . The distance can becomputed using the scalar product information through the equality

Φ(x) − Φ(z)2 = Φ(x),Φ(x)+ Φ(z),Φ(z)− 2Φ(x),Φ(z)= k(x, x) + k(z, z) − 2k(x, z).


11/23

Math Geosci (2009) 41: 397–419 407

Note that this equality is only true for Euclidean distance, hence the necessity of the

MDS procedure prior to performing KKM. For an overview of clustering techniques,

see Buhmann (1995), and Shawe-Taylor and Cristianni (2004) for specific informa-

tion about kernel clustering techniques.

2.4 Uncertainty Quantification

In our application, we apply KPCA or KKM to hundreds of realizations of a spatial

phenomenon mapped in a Euclidean space. These methods identify a small subset of

realizations which represent “typical” realizations of the full set. In Fig. 6(A), we use

the same example as in Fig. 4 using the Hausdorff distance—the subset of realizations

selected by KPCA is represented by squares. Application of the transfer function

(e.g., flow simulation) is then done on this subset of realizations. Uncertainty can be

subsequently analyzed by calculating, for example, the quantiles P10, P50 and P90 onthese few models (Fig. 6(B)). Note that the subset of realizations selected by KPCA

or KKM may not be equiprobable. Thus, a weighting scheme should be defined for

a proper estimation of the quantiles. In the case of KKM, it is obvious to represent

each simulation as many times as the number of models in the corresponding cluster.

In the case of KPCA, we propose to perform k-means in the subspace generated by

KPCA to define the weighting scheme.

2.5 Illustration of the Concept in a Simple Example

Before providing a realistic application of this methodology, we give a simple il-

lustrative example that describes intuitively the inner working of the proposed ap-

proach. In this example shown in Fig. 7, we have generated 30 2D-realizations

xi = (x i1, xi2), i = 1, . . . , 30. Their representation in the 2D parameter space is given

by 3 clusters as shown by the points, colored by cluster. Such clusters could represent

three “geological populations”. The transfer function for this example is defined by

f (x1, x2) = 0.5x1 +x32 . The contours of the values of the transfer function are shown

Fig. 6 (A) Mapping space R: points selected by KPCA represented by squares, (B) Resulting quantiles

estimation (P10 P50 and P90)


12/23

408 Math Geosci (2009) 41: 397–419

Fig. 7 Simple example demonstrating the benefits of using a good distance to group realizations having

similar transfer function values

in Fig. 7(A). Note that in the general case, for CPU reasons, the transfer function can-

not be evaluated exhaustively. To assess response uncertainty on these realizations,

one could select the centroids of each cluster in Fig. 7(A) and evaluate the transfer

function. This would result in 2 evaluations with similar responses, which should beavoided in the case of a CPU demanding transfer function.

However, assume we can define a distance between realizations i and j , which is

defined as a weighted difference between the coordinates of the points (3).

d ij = 0.5

xi1 − xj 1

+

xi2 − xi2

. (3)

The distance is a 1st order approximation of the actual difference in transfer function.

The correlation coefficient between d ij and the actual distance is 0.81 (Fig. 7(B)). If

we compute the distance between each pair of realizations, we can map all the realiza-

tions using MDS into a new 2D space as shown in Fig. 7(C). We can see that 2 groups

are identified by traditional k-means in the MDS space, and only 2 transfer function

evaluations are necessary. The probability density derived from these 2 transfer func-

tion evaluations reproduces accurately the density of the response of the 30 realiza-

tions (Fig. 7(D)). The probability densities were generated using a kernel smoothing


13/23

Math Geosci (2009) 41: 397–419 409

algorithm (Bowman and Azzalini 1997). This example, albeit simple, emphasizes an

important point that applies generally. Spatial model realizations may exist in a high-

dimensional space (e.g., the number of grid cells). Examining the responses in such

a high-dimensional space may be prohibitive. However, certain realizations may ap-

pear different from a spatial/geological point of view but may have similar responses.Instead, if a response specific distance exists, the realizations can be mapped into a

low-dimensional response space (Fig. 7(C)) allowing efficient and effective selection

of representative realizations for quantifying response uncertainty. Note that, in the

example above, the realizations in the response space behave linearly and clustering

techniques can be applied directly in this space. However, in real applications, the

Euclidean space resulting from the MDS procedure is often very complex because of

the high non-linearity of the transfer function. Thus, kernel techniques are employed

to make the clustering procedure more efficient. Results of the application of KPCA

and KKM for an oil reservoir example are presented in next section.

3 Application to Subsurface Flow Uncertainty Assessment

In reservoir engineering, several realizations of petrophysical and/or lithological

models of the subsurface reservoir are generated using a geostatistical algorithm. The

transfer function in this application is a numerical flow simulator, which can be very

CPU demanding for models with a large number of grid cells. The realizations are

submitted to flow simulation in order to assess the uncertainty in reservoir flow per-

formance. We use the method proposed in this paper to perform only a small numberof flow simulations whose response has the same characteristics as the entire set of

realizations. The probability distribution of the flow response of interest, in this case

the field production oil rate, is then determined. A synthetic case study is presented

to demonstrate the potential of the proposed methodology. We consider a channel

system, composed of mud and sand. The background mud is understood as a seal-

ing rock which does not have flow capacity and storage capacity. The inner channel

heterogeneity in petrophysical properties is considered as negligible, thus channels

are modeled with uniform (known) porosity/permeability. The spatial distribution of

channel sands is considered the main driving parameter for flow. The reservoir modelis a 80 × 80 2D grid containing 3 producers and 3 injectors, all penetrating channel

sand. To construct a prior uncertainty space that accommodates a large set of model

realizations, 5 realizations derived from 81 training images (giving a total of 405

facies realizations) are geostatistically simulated. The realizations are conditioned

to facies observations at the wells, depicted in Fig. 8, all penetrating channel sand.

A multipoint geostatistical technique called SIMPAT (Arpat and Caers 2007) is used.

Note that in this case, we vary a key input component of the algorithm, namely the

training image. Since the training image determines the nature of the pattern being

simulated, the 405 realizations exhibit significantly varying patterns. In order to ana-

lyze the efficiency and quality of the method, standard flow simulations were run for

each realization. Note that for real field cases, this is in general not possible. For a

more detailed description of the case, see Suzuki and Caers (2008).

We apply the methodology proposed in this paper using flow-based distances mea-

sured using streamline simulation. The Hausdorff distance showed a poor correlation


14/23

410 Math Geosci (2009) 41: 397–419

Fig. 8 Example of 3 reservoir model realizations with well locations

with the flow simulations, thus was not well suited to the application. We have found

that flow-based distances are well adapted when the response of interest is produc-

tion data, which is the case in this study. Streamline simulation has been shown to be

orders of magnitude faster than standard flow simulation, and is thus well suited for

problems where rapid evaluation of many models is needed (Batycky et al. 1997). Our

results are compared with traditional methods, such as ranking with static properties

and tracer simulations.

3.1 Application of the Proposed Methodology

3.1.1 Construction of the Distance Matrix

The distance is calculated using the tracer simulation option of a commercial stream-

line flow simulator. Tracer simulation approximates the flow as linear, meaning that

the injection and production fluids are assumed to be identical. Thus, in this case,

only a single pressure solve is necessary to perform fluid flow simulation, giving ex-

tremely rapid results. The distance between any two reservoir models is given as the

absolute difference in field oil rate at two given times (10 000 and 20 000 days):

δij =

t ∈{10000,20000}

FOPRstreamlinei (t) − FOPRstreamlinej (t).

The field oil rate for each model differs due to the difference in water breakthrough

for each producing well. Late simulation times are employed to ensure that the water

breakthrough has occurred for all realizations, enabling the greatest distinction in the

water breakthrough between each realization. As discussed before, the distance needs

to be reasonably well correlated with the flow response of interest. To compare the

distance with the results from standard flow simulation, we calculate the average of absolute difference in oil rate as

FOPRij =1

N t

N t t =1

FOPREclipsei (t) − FOPREclipsej (t), (4)


15/23

Math Geosci (2009) 41: 397–419 411

where t represents the time and N t the number of timesteps. In this case, the correla-

tion coefficient between the distance and the difference in oil is ρ(δ,FOPR) = 0.77

which we have found is generally sufficient for accurate results. A smaller correlation

coefficient will not necessarily result in inaccurate uncertainty quantification; most of

the time increasing the number of clusters to retain more simulations is sufficient.

3.1.2 Multi-Dimensional Scaling

Using the dissimilarity distance previously defined, we apply multidimensional scal-

ing to map all the realizations in a 3D Euclidean space R. A 3D mapping space R is

deemed appropriate to ensure that the Euclidean distance between any two points in

R reproduces the dissimilarity in the matrix D. Indeed, the correlation coefficient be-

tween the dissimilarity matrix D and the pair-wise Euclidean distance is high at 0.9.

Subsequent application of KPCA and KKM only considers these Euclidean distancesbetween the models since Euclidean distance is a very good representation of the dis-

similarities of the reservoir models. Note that distinct geological properties give rise

to a wide scatter of flow responses. Note as well that certain realizations may appear

different from a geological point of view, but they may exhibit similar flow behavior.

In this case, these realizations would then be found in the same cluster, regardless

which training image they were derived from.

3.1.3 Application of KPCA

At this step of the methodology, we define a kernel function which transforms the

mapping space R into a space F with improved linear variation. The Gaussian radial

basis kernel is used (2), whose parameter was chosen as 10% of the range of the

dissimilarity matrix: σ = 800. We first perform KPCA to reduce the dimensionality

of the problem by projecting the points in a 4D subspace of the feature space. In

that subspace, we perform cluster analysis using k-means, to determine 15 clusters.

The number of cluster was defined as the maximum number of flow simulations we

can afford for a given CPU. We compute the pre-images of each centroid using the

Schölkopf fixed-point algorithm.Full flow simulations are performed for 15 realizations corresponding to the pre-

images. Recall that the pre-image mapping may not result in points corresponding

to a realization. In this instance, we select the nearest point (realization) for flow

simulation. Uncertainty quantification is then performed by calculating the quantiles

P10, P50 and P90 on these 15 models as a function of time, each model being rep-

resented as many times as the number of models in the corresponding cluster. Thus,

only 15 flow simulations were performed out of a total of 405 reservoir models. Fig-

ure 9(A) represents the evolution of field oil rate as a function of time for the 15

selected realizations; Fig. 9(B) represents the quantiles resulting from the 15 and 405

simulations, respectively. We can observe that the estimation of quantiles P10, P50

and P90 is accurate. In Figs. 9(C) and 9(D), we present the probability density of

the field oil rate for the 405 realizations (dotted line), and for the 15 realizations for

2 different times (2000 and 6000 days). Table 1 shows the mean, variance, kurtosis,

and skewness coefficients of the densities. We can see that the estimated densities are


16/23

412 Math Geosci (2009) 41: 397–419

Fig. 9 KPCA Results: (A) Oil Rate as a function of time for all the 15 realizations, (B) P10, P50 and P90

values, (C) Density of oil rate for all 405 realizations (dashed line) and 15 selected realizations (solid line)

at 2000 days and (D) at 6000 days

Table 1 Statistical properties of the densities resulting from KPCA and KKM

All Data KPCA KKM All Data KPCA KKM

2000 days 2000 days 2000 days 6000 days 6000 days 6000 days

Mean 4.11E+03 4.84E+03 4.35E+03 3.25E+03 3.20E+03 2.63E+03

Variance 2.04E+06 1.01E+06 1.25E+06 4.29E+06 1.53E+06 2.07E+06

Kurtosis 1.7998 1.7998 1.7998 1.7998 1.7998 1.7998

Skewness 8.71E−16 5.51E−16 −2.99E−15 −1.74E−16 −1.13E−15 1.24E−15

close to the reference, which means that the 15 realizations have similar characteris-

tics as the 405. We now replace this two-step methodology (PCA and clustering) by

performing a single cluster analysis in the feature space F .


17/23

Math Geosci (2009) 41: 397–419 413

3.1.4 Application of KKM

In this section, we apply the second methodology, kernel k-means (KKM). The two

first steps, the definition of the dissimilarity matrix and mapping the points with

MDS, are identical for the 2 methods. Thus, they are not illustrated here. A Gaussian

kernel (2) with σ = 850 is used to define the feature space F in which clusters are

identified. Again, we assume that only 15 flow simulations are affordable in this case.

Uncertainty quantification is subsequently performed by flow simulation for the 15

realizations selected by KKM and by computing the resulting quantiles P10, P50 and

P90 (Figs. 10(A) and 10(B)). The weight of each realization equals the number of

points in the corresponding clusters. Figures 10(C) and 10(D) represent the density

computed from the 15 simulations, as well as the density for the full set of realiza-

tions for 2 different times. Table 1 gives their statistical properties. We can see thatthe subset of 15 selected realizations has similar probability densities compared to

the entire set of 405 realizations.

Fig. 10 KKM Results: (A) Oil Rate as a function of the time for all the 15 realizations, (B) P10, P50, P90

values, (C) Density of oil rate for all 405 realizations (dashed line) and 15 selected realizations (solid line)

at 2000 days and (D) at 6000 days


18/23

414 Math Geosci (2009) 41: 397–419

3.2 Comparison with Classical Ranking Techniques

In this section, we compare the results obtained by the proposed methodology with

classical techniques which consists of ranking the realizations according to a specific

measure, and then determining realizations for flow simulation.

3.2.1 Ranking Technique Review

The central goal of ranking is to exploit a relatively simple static measure to accu-

rately select geological realizations that correspond to the targeted percentiles of the

production responses, for example, those which represent P10, P50 and P90 (Ballin et

al. 1992). This would define the bounds of the uncertainty without performing a large

number of fine-scale flow simulations. The ranking and selection of realizations must

be tailored to the flow process. It is well known that a particular ranking measuremust be highly correlated to production response. Conventional ranking measures

are, for example, original oil in place or connectivity (McLennan and Deutsch 2005).

Use of streamline simulation (Gilman et al. 2002) and tracer simulation (Ballin et

al. 1992) have received significant attention. However, there is no unique ranking

index when there are multiple flow response variables and no ranking measure is

perfect.

3.2.2 Comparison of Quantile Estimation

In this work, we have considered two different measures for each of the 405 real-

izations: original oil in place (OOIP) which represents the total volume of oil in the

reservoir, and oil rate obtained by streamline tracer simulation, as used for the dis-

similarity distance. Once a ranking measure is selected, the methodology for ranking

and selecting the geostatistical realizations for flow processing is straightforward.

The ranking measure is calculated for every geostatistical realization. The low (P10),

medium (P50) and high (P90) geological realizations are then selected for flow mod-

eling. Results for oil rate are presented in Fig. 11. We have presented the quantiles

obtained with the entire set of realizations, and the quantiles resulting from the rank-ing measure. Figure 11(A) represents results using OOIP as a ranking measure. Fig-

ures 11(B) to 11(D) represent results using the oil rate from the tracer at respec-

tively 7000, 10 000, and 20 000 days. Quantile estimations using ranking based on

OOIP and streamline tracer are less accurate than the one obtained with the proposed

methodology. To understand why the OOIP and tracer rankings provide less accu-

rate results, we examine the correlation coefficients of the ranking measurement with

the field oil rate. The correlation coefficient between OOIP and field oil rate is 0.19,

which indicates that the OOIP is not a good measure for ranking. Tracer simulation is

more suitable due to the improved correlation with the flow response (0.63, 0.76, and

0.87 at 7000, 10 000, and 20 000 days, respectively). This illustrates that in order for

the ranking procedure to give reliable results, the ranking measure must be strongly

correlated with production.

The ranking methods for selecting the P10, P50 and P90 realizations for flow sim-

ulation are not as accurate as the methodology proposed in this paper. However, only


19/23

Math Geosci (2009) 41: 397–419 415

Fig. 11 Quantiles P10, P50 and P90 resulting from ranking measures: (A) OOIP, (B) Tracer—7000 days,

(C) Tracer—10 000 days, (D) Tracer—20 000 days

3 full flow realizations were performed, whereas for the new method, 15 flow sim-

ulations were necessary. To compare both methods based upon the same number of

flow simulations, we select 15 realizations equally spaced according to the ranking

measure. Resulting quantiles for field oil rate are presented in Fig. 12. For the samenumber of flow simulation, Fig. 12 shows that the use of ranking measures is less

accurate than the use of KPCA or KKM. Note that in the method proposed in this

paper, we use tracer simulations for calculating the distance but not for ranking. The

selection of realizations is done using another approach (KPCA or KKM). In the

case shown here, results show that for the same measure (tracer simulation), better

results are obtained from KPCA or KKM than from ranking (Fig. 13). In Fig. 13,

we observe that the efficiency of the ranking technique relies on a high correlation

coefficient between the ranking measurement and the flow response, whereas in case

of distances, a smaller correlation coefficient is sufficient. In addition, for an equiv-

alent correlation coefficient, quantiles are more accurate using distances and kernels,

than using ranking measures (Fig. 14). In other words, the use of distances, instead

of absolute values, improves the solution by using a “measure” to select the realiza-

tions.


20/23

416 Math Geosci (2009) 41: 397–419

Fig. 12 Quantiles P10, P50 and P90 resulting from ranking measures: (A) OOIP, (B) Tracer—7000 days,

(C) Tracer—10 000 days, (D) Tracer—20 000 days

4 Conclusions

We have presented results for a new realization-based method for uncertainty quan-

tification of a spatial phenomenon. The method relies on a reasonable correlation

between the distance measure and the response variables of interest. Using an ap-

plication specific distance is an important additional tool which makes the task of

response uncertainty quantification more effective. In general, each new type of ap-plication will require investigation of a new distance, which requires more work and

produces more reward. For similar types of problems, these distances can then be

reused. In our example of assessing subsurface flow uncertainty, we use streamline

simulation to obtain the distances, which correlates well with the differences in pro-

duction response using standard flow simulation. Given the distance measure, we

employ KPCA and k-means clustering or kernel k-means to select a subset of 15 re-

alizations which contains the same P10, P50, P90 quantiles as for the entire set of

405 models. The application of this new method shows promising results; quantile

estimations using this methodology are noticeably better than those using traditional

ranking methods for the same number of transfer function evaluations. In addition,

only a small number of transfer function evaluations were necessary to obtain accu-

rate uncertainty quantification though quantile estimation. An application to a simple

synthetic case is provided in this paper. However, this methodology can also be suc-

cessfully applied on oil and gas reservoirs, as presented in Scheidt and Caers (2008),


21/23

Math Geosci (2009) 41: 397–419 417

Fig. 13 Comparison between ranking and KKM using the same tracer measure. Note that the correlation

coefficient for the absolute values of ranking measure is greater than the relative (distance) measure, butthe P10, P50, P90 estimations are less accurate

Fig. 14 Comparison between ranking and KKM for similar correlation coefficient. Note that for similar

correlation coefficient, P10, P50, P90 estimations are more accurate for kernels


22/23

418 Math Geosci (2009) 41: 397–419

where it was shown that KPCA and KKM easily outperform the state of the art rank-

ing technique. Tests on the robustness of the method with regards to the correlation

between the distance and the difference of transfer function evaluations were per-

formed on a real oil field case (Scheidt and Caers 2008). Results show that the higher

the correlation, the smaller the error in the quantile estimation. In addition, if the cor-relation is low, increasing the number of transfer function evaluations is required in

order to obtain accurate representation of uncertainty. However, no systematic bias or

reduction in variance was noted. In the case where the correlation is zero, the method

does not better or worse than random selection.

Acknowledgements The authors would like to acknowledge the SCRF sponsors and Chevron for their

support. Many thanks as well to Darryl Fenwick from StreamSim Technologies for his help using 3DSL,

and for many useful discussions.

References

Arpat BG (2005) Sequential Simulation with patterns. PhD dissertation, Stanford University, USA, 166 p

Arpat BG, Caers J (2007) Stochastic simulation with patterns. Math Geol 39(202):177–203

Ballin PR, Journel AG, Aziz K (1992) Prediction of uncertainty in reservoir performance forecast. JCPT

31(4):52–62

Batycky RP, Blunt MJ, Thiele MR (1997) A 3D field-scale streamline-based reservoir simulator. SPERE

12(4):246–254

Borg I, Groenen P (1997) Modern multidimensional scaling: theory and applications. Springer, New York,

614 pBowman AW, Azzalini A (1997) Applied smoothing techniques for data analysis. Oxford University Press,

London, 204 p

Buhmann JM (1995) Data clustering and learning. In: Arbib M (ed) The handbook of brain theory and

neural networks. MIT Press, Cambridge, pp 278–281

Box GEP, Draper NR (1975) Robust designs. Biometrica 62(2):347–352

Cox TF, Cox MAA (1994) Multidimensional scaling. Chapman and Hall, London, 213 p

Damslet E, Hage A, Volder R (1992) Maximum information at minimum cost! A north field development

study with an experimental design. JPT 44(12):1350–1356

Dubuisson M-P, Jain AK (1994) A modified Hausdorff distance for object matching. In: Proceeding of the

international conference on pattern recognition, Jerusalem, vol A, pp 566–568

Gilman JR, Meng H-Z, Uland MJ, Dzurman PJ, Cosic S (2002) Statistical ranking of stochastic geomodels

using streamline simulation: a field application. In: SPE annual technical conference and exhibition.SPE 77374

Journel A, Alabert F (1990) New method for reservoir mapping. JPT 42(2):212–218

Keerthi SS, Lin C-J (2003) Asymptotic behaviors of support vector machines with Gaussian kernel. Neural

Comput 15(7):1667–1689

Manceau E, Zabalza-Mezghani I, Roggero F (2001) Use of experimental design to make decisions in an

uncertain reservoir environment—from reservoir uncertainties to economic risk analysis. In: OAPEC

conference, Rueil, France, 26–28 June 2001

McLennan JA, Deutsch CV (2005) Ranking geostatistical realizations by measures of connectivity. In:

SPE/PS-CIM/CHOA, international thermal operations and heavy oil symposium, Calgary, Canada.

SPE 98168-MS

Park K, Caers J (2007) History matching in low-dimensional connectivity-vector space. UnpublishedSCRF report 20, Stanford University

Sarma P (2006) Efficient closed-loop optimal control of petroleum reservoirs under uncertainty. PhD dis-

sertation, Stanford University, USA, 201 p

Scheidt C, Caers J (2008) A new method for uncertainty quantification using distances and kernel

methods—application to a deepwater turbidite reservoir. SPE J (submitted)

Schöelkopf B, Smola A (2002) Learning with kernels. MIT Press, Cambridge, 664 p


23/23

Math Geosci (2009) 41: 397–419 419

Schöelkopf B, Smola A, Muller K-R (1998) Nonlinear component analysis as a kernel eigenvalue problem.

Neural Comput 10(5):1299–1318

Shawe-Taylor J, Cristianni N (2004) Kernel methods for pattern analysis. Cambridge University Press,

Cambridge, 462 p

Shi J, Malik J (2000) Normalized-cut and image segmentation. IEEE Trans Pattern Anal Mach Intell

22(8):888–905Suzuki S, Caers J (2008) A distance based prior model parameterization for constraining solution of spatial

inverse problems. Math Geosci 40(4):445–469

Vapnick VN (1998) Statistical learning theory. Wiley, New York, 736 p

scheidt and caers (2010)

Documents